# Vector Fuzzy c-Spherical Shells (VFCSS) over Non-Crisp Numbers for Satellite Imaging

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Definition of Various Non-Crisp Numbers

#### 2.2. Various Metrics for Symbolic Numbers

#### 2.3. Various Metrics for Fuzzy Numbers

#### 2.4. VFCSS Applied to Crisp Numbers

#### 2.5. The VFCSS Applied to Fuzzy Numbers

#### 2.5.1. The VFCSS Applied to LR-Type Fuzzy Numbers

#### 2.5.2. The VFCSS Applied to TFN-Type Fuzzy Numbers

#### 2.5.3. The VFCSS Applied to TAN-Type Fuzzy Numbers

#### 2.6. The VFCSS over Symbolic Numbers

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Proposed VFCSS clustering over input LR-type fuzzy numbers include four interference classes red, blue, black and green with (20,20,20,20) members, (first step of state 1).

**Figure 3.**Proposed VFCSS clustering over input LR-type fuzzy numbers include four complexity interference classes red, blue, black and green with (20,20,20,20) members, (second step of state 1).

**Figure 4.**Proposed VFCSS clustering over input non-symmetric LR-type fuzzy numbers include four interference classes red, blue, black and green with (20,20,20,20) members, (first step of state 2).

**Figure 5.**Proposed VFCSS clustering over input non-symmetric LR-type fuzzy numbers include four complexity interference classes red, blue, black and green with (20,20,20,20) members (second step of state 2).

**Figure 6.**Conventional FCSS clustering over crisp input numbers include four interference classes red, blue, black, and green with (20,20,20,20) members, (first step of state 3).

**Figure 7.**Conventional FCSS clustering over crisp input numbers include four complexity interference classes red, blue, black and green with (20,20,20,20) members (second step of state 3).

**Figure 8.**Three georeferenced satellite images (obtained using Google maps) obtained over three different dates (

**a**) 23-10-2011 (

**b**) 09-09-2016 (

**c**) 30-12-2018 (used for state 4).

**Figure 9.**The extracted circles from images (a), (b) and (c) Figure 8 represented by red, green and blue colors, respectively.

**Figure 10.**A fish farm satellite image (

**Left**) and the extracted circles from the image and radii results (

**Right**) using VFCSS algorithm.

d${t}_{i}$ | ${v}_{i,j},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}=Norm\left(\raisebox{1ex}{$\partial {w}_{i}^{T}$}\!\left/ \!\raisebox{-1ex}{$\partial {t}_{i}$}\right.\right)=\raisebox{1ex}{$\partial {w}_{i}^{T}$}\!\left/ \!\raisebox{-1ex}{$\partial {t}_{i}$}\right.$ | $\left[\stackrel{\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

**Table 2.**$\raisebox{1ex}{$\partial {w}^{T}$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.$ for VFCSS applying over LR-type fuzzy numbers when metric (13) is used.

${t}_{i}$ | ${m}_{1{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${m}_{2{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\alpha}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\beta}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

$\raisebox{1ex}{$\partial {w}_{i}^{T}$}\!\left/ \!\raisebox{-1ex}{$\partial {t}_{i}$}\right.$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,0,1,0,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},0,1,0,1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}},-l,\stackrel{4\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+3}{\stackrel{\u23de}{0,0,\cdots ,0}},r,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${m}_{1{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${m}_{2{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\alpha}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\beta}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,0,1,0,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},0,1,0,1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+3}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${m}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\alpha}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\beta}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{3\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,1,1,\stackrel{3\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{3\left(j-1\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{3\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{3\left(j-1\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{3\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${v}_{1{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{2{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{3{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{4{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,1,1,1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},-1,1,-1,1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+3}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${v}_{1{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{2{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{3{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${v}_{4{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{4\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)+3}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+2}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{4\left(j-1\right)+3}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{4\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${v}_{i,j}^{L},forj=1,2,\cdots ,p$ | ${v}_{i,j}^{C},forj=1,2,\cdots ,p$ | ${v}_{i,j}^{R},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{5\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,00,1,0,\stackrel{5\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{5\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},0,1,0,\sqrt{\frac{{\alpha}_{1}}{{\alpha}_{1}+6{\alpha}_{2}+{\alpha}_{3}}},\sqrt{\frac{{\alpha}_{3}}{{\alpha}_{1}+6{\alpha}_{2}+{\alpha}_{3}}},\stackrel{5\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{5\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},0,0,1,0,1,\stackrel{5\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

${t}_{i}$ | ${\alpha}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ | ${\beta}_{{\tilde{v}}_{i,j}},forj=1,2,\cdots ,p$ |

${\xi}_{{t}_{i}}$ | $\left[\stackrel{2\left(j-1\right)}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{2\left(p-j\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ | $\left[\stackrel{2\left(j-1\right)+1}{\stackrel{\u23de}{0,0,\cdots ,0}},1,\stackrel{2\left(p-j\right)}{\stackrel{\u23de}{0,0,\cdots ,0}}\right]$ |

Step | First Step | Second Step | |
---|---|---|---|

Output Clusters | |||

Confusion Matrix | $\left[\begin{array}{c}20,0,0,0\\ 1,18,0,1\\ 0,0,20,0\\ 0,1,0,19\end{array}\right]$ | $\left[\begin{array}{c}20,0,0,0\\ 0,20,0,0\\ 2,1,16,1\\ 0,1,1,18\end{array}\right]$ |

Step | First Step | Second Step | |
---|---|---|---|

Output Clusters | |||

Confusion Matrix | $\left[\begin{array}{c}18,1,0,1\\ 0,19,0,1\\ 0,0,20,0\\ 0,0,0,20\end{array}\right]$ | $\left[\begin{array}{c}20,0,0,0\\ 0,19,0,1\\ 1,2,15,2\\ 0,1,2,17\end{array}\right]$ |

Step | First Step | Second Step | |
---|---|---|---|

Output Clusters | |||

Confusion Matrix | $\left[\begin{array}{c}20,0,0,0\\ 1,18,0,1\\ 0,0,20,0\\ 0,1,0,19\end{array}\right]$ | $\left[\begin{array}{c}8,7,4,1\\ 7,8,3,2\\ 1,0,19,0\\ 0,1,1,18\end{array}\right]$ |

**Table 12.**Simulation results for state 4 and comparison of resulting radii with FCSS and VFCSS algorithms.

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**MDPI and ACS Style**

Abaspur Kazerouni, I.; Mahdipour, H.; Dooly, G.; Toal, D.
Vector Fuzzy c-Spherical Shells (VFCSS) over Non-Crisp Numbers for Satellite Imaging. *Remote Sens.* **2021**, *13*, 4482.
https://doi.org/10.3390/rs13214482

**AMA Style**

Abaspur Kazerouni I, Mahdipour H, Dooly G, Toal D.
Vector Fuzzy c-Spherical Shells (VFCSS) over Non-Crisp Numbers for Satellite Imaging. *Remote Sensing*. 2021; 13(21):4482.
https://doi.org/10.3390/rs13214482

**Chicago/Turabian Style**

Abaspur Kazerouni, Iman, Hadi Mahdipour, Gerard Dooly, and Daniel Toal.
2021. "Vector Fuzzy c-Spherical Shells (VFCSS) over Non-Crisp Numbers for Satellite Imaging" *Remote Sensing* 13, no. 21: 4482.
https://doi.org/10.3390/rs13214482